This invention relates to a teaching method and apparatus for calculus and fluid flow classes comprising the use of fluid flow through a plurality of variously configured funnels in order to physically demonstrate physical principles of fluid flow and the mathematical principles of integration.
Math anxiety, or fear of mathematics, is perhaps nowhere more evident than in the calculus classroom. Calculus, with its incremental iterations, is one of the great stumbling points of mathematical conceptual development, standing alongside division, the equal sign, and modern algebra in elusiveness. Calculus students, typically 17-20 years of age, are principally concrete thinkers (including so-called horizontal decalage of formal thinkers), who are persons who have difficulty learning mathematical principles in the first instance in terms of abstract symbols representing variables. In more formal terms, such calculus students lack equilibrated cognitive structures-they fail to account for multiple situational possibilities. For concrete thinkers, learning mathematical principles in terms of abstract symbols is facilitated if the principles are first demonstrated by a model or concrete example. Again, in more formal terms, concrete particulars are propaedeutic to incipient learning cycles foundational to semeiotic or symbolic abstraction; that is, only after having established a functionally assimilative cognitive scheme via motoric activities are concrete thinkers able to proceed to semeiotic empirical generalization. Without a functionally assimilative cognitive scheme, conceptual verbalizations tend to be idiosyncratically interpreted by each student, rendering purely semeiotic exposition vague and ambiguous and learning sporadic and superficial. Nonetheless, calculus, elusory as it is, is principally taught semeiotically (symbolically), approaching students as though formal thinkers, able to exhaustively appreciate manifold situational variables and spontaneously ratiocinate hypotheses as to eventualities, which chiefly they are unable to do, though origins of the symbolic function are acquired at the sensorimotor stage. Thus high school and college students of calculus often grapple with the semeiotically expounded infinitesimal iterations of differential and integral calculus in vain, emerging unable to apply the rotely memorized procedures to concrete problems and confused. It is one of our biological inheritances that confusion produces emergency anxiety, along with the attendant defensive measures of flight, fright, and freezing. Hence, the etiology of math anxiety.
The presently available art in calculus teaching aids is comprised of a segmented cone statically illustrating the conic sections, a planar panel upon which points and line segments may be joined into static three-dimensional figures, and static geometric shapes generated by a revolution of an area bounded by a curve which may be serially mounted upon support braces; each of these teaching aids is wholly visual and meant to be handled solely by the teacher. The essentially concretely thinking students are relegated to passively cognizing various shapes exclusively through their visual perceptual fields. The present aids offer no channels of learning through concrete manual manipulation.
Projects involving serial task accomplishment and problem solving are presently being integrated into school system curriculums. The leap from rote learning to noetic fluency is much like the difference in language use between a listener and speaker. The rotely learning student, deficient in organizational persistence and unbound by prior constraints, resorts to episodic empiricism, constructing hypotheses in potshot manner without justification and with no differentiation of the passively received information. In contrast, projects allow the student to put things together for himself, leading him to discovery. Through discoveries, the student constructs hypotheses via cumulative constructionism, wherein the student discovers regularities and relatedness, avoids information drift, and locates constraints giving shape to an hypothesis. The principal problem of human memory is not storage but retrieval, not recognition but spontaneous recall. A given project encourages the student to recast a difficulty into a form that he knows how to work with, imposing a workable form upon the difficulty, to transform information related to the task for immanent transmittal, resulting in information differentiation and reducing the aggregate subjective complexity by imbedding it into a cognitive process or scheme the student has constructed for himself, thereby making the material more accessible for retrieval. The very attitudes and activities that characterize figuring out or discovering things for oneself also conserve memory. The key to retrieval is organization, and good organization is achieved by problem solving. Discovery learning further engenders competence motivation, where growth and maintenance of mastery become central and dominant, whereas extrinsically motivated students become overachievers, lacking capacity for transforming learning into viable thought structures and analytic ability. One learns the working heuristics of discovery only through the effort of inquiry, the exercise of problem solving, and the completion of projects.
A further shortcoming of the presently available calculus teaching aids is that they do not lend themselves to projects of serial tasks and problem solving; they are limited to purely demonstrational teaching strategies by the teacher. That is, the present aids are not adapted to presenting students with a project to be accomplished, and are therefore of little utility to school systems developing curriculums of projects.
Joint effort in classrooms by student groups upon tasks and problems provides for interpersonal interactions between the students. In such interactions, views are limned, questioned, and defended, leading to argumentation, opinions being justified, and thought being clarified. Language thus serves to internalize the resultant views into a compact experiential category. The students are forced to become more coherent, limpid, and logical as a result of joint classroom effort.
The prior calculus teaching aids are limited to teaching strategies wherein the teacher is the exclusive personality manipulating and expounding upon the aids. The presently available teaching aids are inapt for offering avenues of teaching strategies involving interaction or involvement between students.
Another drawback of the presently available art in calculus teaching aids is that it deals exclusively in illustrating static geometric shapes, curves, and surfaces. Yet calculus is uniquely adapted to deal with dynamic quantities and variating factors. The available calculus teaching aids illustrate forms, quite immobile, and are useless in demonstrating the proprieties of the calculus in dealing with time-dependent variation and ensuing transitory states.
A further shortcoming of presently available calculus teaching aids is that derived results of integration are not quantitatively testable. The static forms simply illustrate how to qualitatively set up an integral, but are unsuitable to empirically verify calculated results. Hence students are uncertain about the tenuous validity of the theory and hesitant of the accuracy of their own calculations.